the second fundamental form
The first fundamental form told the ant on the surface about distances and angles measured from inside. But it says nothing about whether the surface bulges out like a dome, dips in like a bowl, or curves up one way and down the other like a saddle — that is information about how the surface bends INTO the surrounding space, and you need to look at the third dimension to see it. The second fundamental form is precisely the instrument that captures this bending.
Here is how it is read. At a point p, take a unit tangent direction and travel along a curve on the surface in that direction. As you move, the surface's unit normal N tilts; equivalently the curve must accelerate to stay on the surface. The second fundamental form II(w, w) measures, for a tangent direction w, how much the surface pulls away from its own tangent plane in the normal direction — to second order, the surface lies at height (1/2) II(w, w) above the tangent plane after a unit step in direction w. In a parametrization x(u, v) with unit normal N, its three coefficients are e = x_uu . N, f = x_uv . N, g = x_vv . N (the normal components of the surface's second derivatives), so II = e du^2 + 2f du dv + g dv^2. Equivalently, II(w, w) = -dN(w) . w, tying it directly to the Gauss map: it records how the normal turns as you move in direction w.
The second fundamental form is the source of all the bending curvatures. Feeding a unit tangent direction w into it gives the normal curvature in that direction; the largest and smallest values as w sweeps around are the principal curvatures k_1 and k_2; their product is the Gaussian curvature K and their average the mean curvature H. A vital caution that separates this from the first fundamental form: the second fundamental form is EXTRINSIC. It changes if you flip the chosen normal (its sign reverses), and the ant cannot measure it from inside — only by reference to the ambient space. The deep surprise (Theorema Egregium) is that although e, f, g themselves are extrinsic, the particular combination K = (e*g - f^2)/(E*G - F^2) turns out to be intrinsic after all.
On a sphere of radius R with the outward normal, every direction bends the same way, and one finds II = (1/R)(E du^2 + 2F du dv + G dv^2) — the second fundamental form is just (1/R) times the first. This says the normal curvature is 1/R in EVERY direction, so the principal curvatures are both 1/R, and the Gaussian curvature is K = (1/R)(1/R) = 1/R^2. On a plane, all second derivatives have zero normal component, so e = f = g = 0, II is identically zero, and the plane has no bending at all.
On a sphere II is (1/R) times I; on a plane II vanishes entirely.
Unlike the first fundamental form, the second fundamental form is extrinsic and sign-dependent: reverse the chosen unit normal and every coefficient flips sign. So II by itself is not a property the surface 'has' intrinsically; only sign-free combinations of it (notably the Gaussian curvature) can be intrinsic.